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  "body": "\n## Definition\n\nLet $X$ be a [[topological space|0CQHFPGJ45]], and let $\\mathcal{F}$ be a [[sheaf|TB6TM6R4MD]] of [[abelian groups|082GHNPJ53]] on $X$.\nLet $\\mathfrak{U} = (U_i)_{i \\in I}$ be an [[open cover|049PXYHW1T]] of $X$ with $I$ a well-ordered set.\nFor $i_0, \\dots, i_p \\in I$ write $U_{i_0 \\dots i_p} = U_{i_0} \\cap \\dots \\cap U_{i_p}$.\nThe *Čech complex* is the [[cochain complex|WQ82C0M3QB]] with\n$$C^p(\\mathfrak{U}, \\mathcal{F}) = \\prod_{i_0 \u003c \\dots \u003c i_p} \\mathcal{F}(U_{i_0 \\dots i_p})$$\nand differential $d \\colon C^p \\to C^{p+1}$ given by\n$$(d\\alpha)_{i_0 \\dots i_{p+1}} = \\sum_{j=0}^{p+1} (-1)^j \\, \\alpha_{i_0 \\dots \\widehat{i_j} \\dots i_{p+1}}|_{U_{i_0 \\dots i_{p+1}}},$$\nwhere the hat means that the index is omitted.\nThe *Čech cohomology* of $\\mathcal{F}$ with respect to $\\mathfrak{U}$ is $\\check{H}^p(\\mathfrak{U}, \\mathcal{F}) = H^p(C^\\bullet(\\mathfrak{U}, \\mathcal{F}))$.\nThe Čech cohomology of $X$ with coefficients in $\\mathcal{F}$ is the direct limit $\\check{H}^p(X, \\mathcal{F}) = \\varinjlim_{\\mathfrak{U}} \\check{H}^p(\\mathfrak{U}, \\mathcal{F})$ over refinements of covers.\n\n## Remarks\n\nThe sheaf axioms give $\\check{H}^0(\\mathfrak{U}, \\mathcal{F}) = \\mathcal{F}(X)$ for every cover.\n\nExample: let $X = S^1$, covered by two open arcs $U_0, U_1$ whose intersection has two components, and let $\\mathcal{F} = \\underline{\\Z}$ be the constant sheaf.\nThen $C^0 = \\Z^2$, $C^1 = \\underline{\\Z}(U_{01}) = \\Z^2$, and $d(a, b) = (b - a, b - a)$.\nSo $\\check{H}^0 = \\Z$ and $\\check{H}^1 = \\Z^2 / \\set{(c, c)} \\cong \\Z$.\n\nNon-example: a family of open sets that does not cover $X$ is not allowed; then $\\check{H}^0$ would be the sections over the union, not over $X$.\n\nThere is a natural map $\\check{H}^p(\\mathfrak{U}, \\mathcal{F}) \\to H^p(X, \\mathcal{F})$ to [[sheaf cohomology|PQ6Q143ARJ]].\nIt is an isomorphism when $H^q(U_{i_0 \\dots i_p}, \\mathcal{F}) = 0$ for all $q \u003e 0$ and all finite intersections (Leray's theorem), for example for a quasi-coherent sheaf on a separated scheme and an affine open cover.\n",
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